In this work, we aim to understand how Arctic sea ice influences the weakening of the large-scale ocean circulation in the Atlantic under future anthropogenic greenhouse gas emissions. We find that when Arctic sea ice and the ocean circulation are allowed to mutually influence each other in a positive feedback cycle, the level of circulation slowdown is highly sensitive to the rate of greenhouse gas increase change (even when the total final amount of greenhouse gas increase is the same). We also show that the retreat of sea ice under global warming can counteract some of the ocean circulation's weakening induced by greenhouse gas increases. This work highlights how understanding the fundamental mechanisms of ocean-sea ice interactions is critical for constraining the future weakening of the Earth's large-scale ocean circulation.